Question

In the expansion of $(1+x+x^2)e^{-x}$, the coefficient of $x^2$ is

• A. $1$
• B. $-1$
• C. $\frac{1}{2}$
• D. $-\frac{1}{2}$

Answer 1

The correct option is C

$\frac{1}{2}$

Explanation for the correct option:

Finding the coefficient of $x^2$ :

Expanding $e^{-x}$ in the given expression

$$\begin{gathered} \left(1+x+x^2\right)\left(1–\frac{x}{1!}+\frac{x^2}{2!}–\frac{x^3}{3!}+\dots .\right) \\ \Rightarrow \left(1–\frac{x}{1!}+\frac{x^2}{2!}–\frac{x^3}{3!}+\dots \right)+\left(x–\frac{x^2}{1!}+\frac{x^3}{2!}–\frac{x^4}{3!}+\dots \right)+\left(x^2–\frac{x^3}{1!}+\frac{x^4}{2!}–\frac{x^5}{3!}+\dots \right) \\ \Rightarrow \left(\frac{x^2}{2!}–\frac{x^2}{1!}+x^2\right)+\left(1–\frac{x}{1!}–\frac{x^3}{3!}+\dots \right)+\left(x+\frac{x^3}{2!}–\frac{x^4}{3!}+\dots \right)+\left(–\frac{x^3}{1!}+\frac{x^4}{2!}–\frac{x^5}{3!}+\dots \right) \end{gathered}$$

As we have to find the coefficient of $x^2$ , we can neglect the other terms

$$\begin{gathered} \Rightarrow \left(\frac{x^2}{2!}–\frac{x^2}{1!}+x^2\right) \\ \Rightarrow \left(\frac{1}{2}–1+1\right)x^2 \\ \Rightarrow \frac{1}{2}{x}^2 \end{gathered}$$

Hence, option (C) is the correct answer.